Question

If the coordinates of a point An are (2^n,2^n +1) then (A1A5)^2 is equal to ?

Answer 1

Task. The coordinates of a point $A_{n}$ are $(2^{n},2^{n}+1)$. Find $|A_{1}A_{5}|^{2}$.

Solution. The square of the distance between points $P(x,y)$ and $Q(x_{1},y_{1})$ is given by the formula:

$|PQ|^{2}=(x-x_{1})^{2}+(y-y_{1})^{2}.$

In our case

$A_{1}(2^{1},2^{1}+1)=(2,3),\qquad A_{5}(2^{5},2^{5}+1)=(32,33),$

whence

$|A_{1}A_{5}|^{2}=(32-2)^{2}+(33-3)^{2}=30^{2}+30^{2}=900+900=1800.$

Answer. 1800.